The Problem With Integer Programming

The Problem with The Problem with Integer ProgrammingInteger Programming

H.P.WilliamsH.P.Williams

London School of EconomicsLondon School of Economics

The Nature of Integer Programming The Nature of Integer Programming (IP)(IP)

Is IP like Linear Programming (LP) ?Is IP like Linear Programming (LP) ?

Applications of Integer Programming Applications of Integer Programming

Mathematical Properties of IPMathematical Properties of IP

Economic Properties of IPEconomic Properties of IP

ChvChváátal Functions and Integer Monoidstal Functions and Integer Monoids

A General (Mixed) Integer A General (Mixed) Integer Programme (IP)Programme (IP)

Maximise/Minimise ∑Maximise/Minimise ∑ jj c cjjxxjj+∑+∑kkddkkyykk

Subject toSubject to :: ∑∑jjaaijijxxj j +∑+∑kkeeikikyyk k <=> b<=> bii for all i for all i

xxjj>=0 all j, y>=0 all j, ykk>=0 all k and integer>=0 all k and integer

A General (Mixed) Integer A General (Mixed) Integer Programme Programme

Frequently (but not always) the integer variables Frequently (but not always) the integer variables are restricted to values 0 and 1 representing are restricted to values 0 and 1 representing (indivisible) Yes/No decisions eg. Investment (indivisible) Yes/No decisions eg. Investment

Can view as a Can view as a LogicalLogical statement about a series statement about a series of Linear Programmes (LPs)of Linear Programmes (LPs)

Leads a to close relationship between Leads a to close relationship between Logic and Logic and IPIP

0-1 Integer Programmes0-1 Integer Programmes

Any IP with bounded integer variables can Any IP with bounded integer variables can be converted to a 0-1 IPbe converted to a 0-1 IP

0-1 IPs can be interpreted as 0-1 IPs can be interpreted as DisjunctionsDisjunctions ofof LPsLPs

Application of Application of logical logical methods to methods to formulation and solutionformulation and solution

Applications of IPApplications of IP Extensions to LPs eg Manufacturing, Distribution, Petroleum, Gas and ChemicalsExtensions to LPs eg Manufacturing, Distribution, Petroleum, Gas and Chemicals

Global Optimisation of non-convex (non-linear) modelsGlobal Optimisation of non-convex (non-linear) models Power Systems LoadingPower Systems Loading Facilities LocationFacilities Location RoutingRouting TelecommunicationsTelecommunications Medical Radiation Medical Radiation Statistical Design Statistical Design Molecular BiologyMolecular Biology Genome SequencingGenome Sequencing Archaeological SeriationArchaeological Seriation Optimal Logical StatementsOptimal Logical Statements Computer DesignComputer Design Aircraft SchedulingAircraft Scheduling Crew RosteringCrew Rostering

Linear Programming v Integer Linear Programming v Integer ProgrammingProgramming

An LPAn LPMinimise XMinimise X22

Subject to: 2XSubject to: 2X11+X+X2 2 >=13>=13

5X5X11+2X+2X22<=30<=30

-X-X11+X+X2 2 >=5>=5

XX11 , X , X22 >= 0 >= 0

The SolutionThe Solution XX11 = 2 = 222

/3/3 , X , X22 = 7 = 722/3/3




5X5X11+2X+2X22<=30<=30

-X-X11+X+X2 2 >=5>=5

XX11 , X , X22 >= 0 >= 0


/3/3 , X , X22 = 7 = 722/3/3

An IPAn IPMinimise XMinimise X22

Subject to: 2XSubject to: 2X11 + X + X2 2 >=13>=13

5X5X11 + 2X + 2X22<=30<=30

-X-X11 + X + X2 2 >=5>=5

XX11 , X , X22 >= 0 and integer >= 0 and integer




5X5X11+2X+2X22<=30<=30

-X-X11+X+X2 2 >=5>=5

XX11 , X , X22 >= 0 >= 0


/3/3 , X , X22 = 7 = 722/3/3

The SolutionThe Solution XX11 = 2 , X = 2 , X22 = 9 = 9

An IPAn IPMinimise XMinimise X22

Subject to: 2XSubject to: 2X11 + X + X2 2 >=13>=13

5X5X11 + 2X + 2X22<=30<=30

-X-X11 + X + X2 2 >=5>=5

XX11 , X , X22 >= 0 and integer >= 0 and integer

LP and IP SolutionsLP and IP Solutions

9 . . . . . 9 . . . . . Min xMin x22

cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30

xx22 -x -x11 + x + x22 >= 5 >= 5

7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0

6 . . . . .6 . . . . .

5 . . . . . 5 . . . . .

0 1 2 3 4 0 1 2 3 4 xx11

LP and IP SolutionsLP and IP Solutions

9 9 Optimal IP Solution (2 , 9)Optimal IP Solution (2 , 9) .. Min xMin x22

cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30

Optimal LP Solution (2 Optimal LP Solution (2 22//33 , 7 , 7 22//33) -x) -x 11 + x + x22 >= 5 >= 5

7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0 xx22

6 . . . . .6 . . . . .

5 . . . . . 5 . . . . .

0 1 2 3 4 0 1 2 3 4 xx11

•

IP Solution after removingIP Solution after removing constraint 1constraint 1

Min xMin x22

8 8 c1c1 . . . . . . cc3 3 stst 5x5x11 + 2x + 2x22 <= 30 <= 30

-x-x11 + x + x22 >= 5 >= 5

. . . . . . . . x x11 , x , x2 2 >= 0 >= 0

xx22 77 . . . . . .

6 . . . . . c6 . . . . . c2 2

Optimal IP Solution (0 , 5)Optimal IP Solution (0 , 5) 5 5

0 1 2 3 4 0 1 2 3 4 xx11

•

IP SolutionIP Solution


cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30


7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0 xx22

6 . . . . .6 . . . . .

5 . . . . . 5 . . . . .

0 1 2 3 4 0 1 2 3 4 xx11

•

IP Solution after removing IP Solution after removing constraint 2constraint 2

9 . . . 9 . . . Min xMin x22

c1 c3 c1 c3 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . . . .. Optimal IP Solution (3, 8)Optimal IP Solution (3, 8) -x-x11 + x + x22 >= 5 >= 5

7 . . . . . 7 . . . . . x x11 , x , x2 2 >= 0 >= 0

xx22

6 . . . . .6 . . . . .

5 . . . . . 5 . . . . . xx11

0 1 2 3 4 0 1 2 3 4

•



cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30


7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0 xx22

6 . . . . .6 . . . . .

5 . . . . . 5 . . . . . xx11

0 1 2 3 4 0 1 2 3 4

•

IP Solution after removing IP Solution after removing constraint 3constraint 3

9 . . 9 . . . . . . Min xMin x22

st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . . . . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30

c1 c2 c1 c2 x x11 , x , x2 2 >= 0 >= 0

7 . . . . . 7 . . . . . xx22

6 . . . . .6 . . . . .

5 . . . . 5 . . . . ..Optimal IP Solution (4 , 5)Optimal IP Solution (4 , 5)

0 1 2 3 4 0 1 2 3 4 xx11

Rounding often not satisfactoryRounding often not satisfactoryExample: The Alabama ParadoxExample: The Alabama Paradox

State Population Fair solution Rounded SolutionState Population Fair solution Rounded Solution With 10 RepresentativesWith 10 Representatives

A 621k 4.41 4A 621k 4.41 4 B 587k 4.17 4B 587k 4.17 4 C 201k 1.43 2C 201k 1.43 2

With 11 RepresentativesWith 11 Representatives A 621k 4.85 5A 621k 4.85 5 B 587k 4.58 5B 587k 4.58 5 C 201k 1.57 1C 201k 1.57 1

IP Formulation of Political IP Formulation of Political Apportionment ProblemApportionment Problem

VVii = Population (Votes cast) for State (Party) i = Population (Votes cast) for State (Party) i xxi i = Seats allotted to State (Party) i= Seats allotted to State (Party) i

Choose xChoose xi i so as to:so as to:

Min MaxMin Maxi i (x(xii / v / vii)) st ∑st ∑iixxi i = Total Number of Seats= Total Number of Seats xxi i >= 0 and integer for all I>= 0 and integer for all I

ie Min yie Min y st xst xii / v / vi i <= y for all i<= y for all i ∑ ∑ ii x xi i = Total Number of Seats= Total Number of Seats xxi i >= 0 and integer for all I>= 0 and integer for all I

LP Relaxation gives fractional solutionLP Relaxation gives fractional solution IP Solution give Jefferson/D’Hondt solutionIP Solution give Jefferson/D’Hondt solution

IP SolutionIP Solution State Population Fractional Rounded Jefferson/ State Population Fractional Rounded Jefferson/ solution solution D’Hondt solutionsolution solution D’Hondt solution (LP) (IP)(LP) (IP)

With 10 RepresentativesWith 10 Representatives A 621k 4.41 4 5A 621k 4.41 4 5 B 587k 4.17 4 4B 587k 4.17 4 4 C 201k 1.43 2 1C 201k 1.43 2 1

With 11 RepresentativesWith 11 Representatives A 621k 4.85 5 5 A 621k 4.85 5 5 B 587k 4.58 5 5B 587k 4.58 5 5 C 201k 1.57 1 1C 201k 1.57 1 1

Mathematical Differences between Mathematical Differences between LP and IPLP and IP

Consider a (Pure) IP in standard formConsider a (Pure) IP in standard form

Maximise cMaximise c11xx11 + c+ c22xx22 + … + c + … + cnnxxnn

Subject to: aSubject to: a1111xx11 + a + a1212xx22 + … a + … a1n1nxxnn <= b <= b11

aa2121xx11 + a + a2222xx22 + … a + … a2n2nxxnn <= b <= b22

.. . . aam1m1xx11+ a+ am2m2xx22 + … a + … amnmnxxnn <= b <= bmm

xx11 , x , x22 , … , x , … , xnn >= 0 and integer >= 0 and integer

Mathematical Differences between Mathematical Differences between LP and IPLP and IP

LP IPLP IP

If has Optimal Solution No limit on number of positive variablesIf has Optimal Solution No limit on number of positive variables there is one with at there is one with at most most mm variables positive variables positive (a (a basic basic solution) solution) Hilbert BasisHilbert Basis (no fixed dimension) (no fixed dimension)

At most At most nn constraints At most constraints At most 22nn – 1– 1

binding at optimum constraints binding binding at optimum constraints binding at optimumat optimum

There are valuations There are valuations ChvChváátaltal Functions Functions on constraints which on constraints which close duality gap ieclose duality gap ie there is a (symmetric) LP No obvious symmetrythere is a (symmetric) LP No obvious symmetry (dual) model(dual) model

IPs involve IPs involve Lattices Lattices within within PolytopesPolytopes

EgEg

Max 2xMax 2x11+x+x22

st 2xst 2x11+9x+9x22<=80 <=80

2x2x11-3x-3x22<=6<=6 -x-x11 <=0 <=0 -x-x22<=0<=0 2x2x11+3x+3x2 2 ≡0(mod12) ≡0(mod12) xx1 1 ≡0(mod1) ≡0(mod1) xx22≡0(mod1)≡0(mod1)

What are the strongest What are the strongest implications? implications?

Max 2xMax 2x11+x+x2 2

st 2xst 2x11+9x+9x22<=80 2x<=80 2x11+9x+9x22<=80 <=80

2x2x11-3x-3x22<=6 2x<=6 2x11-3x-3x22<=6 <=6 -x-x11 <=0 -x <=0 -x11 <=0 2x <=0 2x11+x+x2 2 ? ? -x-x22<=0 -x<=0 -x22<=0 <=0

2x2x11+3x+3x2 2 ≡0(mod12) 2x≡0(mod12) 2x11+3x+3x2 2 ≡0(mod12) ≡0(mod12) xx1 1 ≡0(mod1) x ≡0(mod1) x1 1 ≡0(mod1) 2x ≡0(mod1) 2x11+x+x2 2 ? ?

xx22≡0(mod1) x≡0(mod1) x22 ≡0(mod1) ≡0(mod1)

What are the strongest What are the strongest implications? Dual arguments.implications? Dual arguments.

Max 2xMax 2x11+x+x2 2

st 2xst 2x11+9x+9x22<=80 2x<=80 2x11+9x+9x22<=80<=80 ⅓⅓

2x2x11-3x-3x22<=6 2x<=6 2x11-3x-3x22<=6<=6 ⅔⅔ -x-x11 <=0 -x <=0 -x11 <=0 <=0 00 2x2x11+x+x2 2 <= 30<= 3022

/3/3 -x-x22<=0 -x<=0 -x22<=0<=0 00

2x2x11+3x+3x2 2 ≡0(mod12) 2x≡0(mod12) 2x11+3x+3x2 2 ≡0(mod12)≡0(mod12) 33 xx1 1 ≡0(mod1) x ≡0(mod1) x1 1 ≡0(mod1) ≡0(mod1) 00 2x2x11+x+x2 2 ΞΞ0(mod4)0(mod4)

xx22≡0(mod1) x≡0(mod1) x22 ≡0(mod1) ≡0(mod1) 4 4


88 .. . .. .

.. .. Objective = 30Objective = 3022//33

44 .. .. . . Objective = 28 Objective = 28 .. . .

.. . . . . Objective = 24 Objective = 24 0 6 120 6 12


Optimisation over Optimisation over polytopespolytopes give strongest (LP) bound on give strongest (LP) bound on objectiveobjective

Optimisation over Optimisation over latticeslattices give strongest congruence give strongest congruence relation for objectiverelation for objective

Combined they give rank 1 cut for objectiveCombined they give rank 1 cut for objective

This may not be adequateThis may not be adequate

Lattices Lattices within within Cones Cones give give Integer Integer MonoidsMonoids

These are a fundamental structure for IPThese are a fundamental structure for IP

Polyhedral and Non-Polyhedral Polyhedral and Non-Polyhedral MonoidsMonoids

The integer lattice within the polytope -2x + 7y >= 0The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0x – 3y >= 0A Polyhedral MonoidA Polyhedral Monoidyy

44 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .33 . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .22 . . . . . . . . . . . . .. .. . . . . . . . . . . . . . .1 1 . . . . . . .. . . . . . . . . . . . ……. . . . . . . . . . . . …….00 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x

Projection: A non-polyhedral monoid (Generators 3 and 7)Projection: A non-polyhedral monoid (Generators 3 and 7)

x x .. . . x x .. .. x x x x . . x x x x .. x x x x x x…….…….


44 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .33 . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .22 . . . . . . . . . . . . .. .. . . . . . . . . . . . . . .1 1 . . . . . . .. . . . . . . . . . . . ……. . . . . . . . . . . . …….00 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x

Projection: A non-polyhedral monoid (Generators 3 and 7)Projection: A non-polyhedral monoid (Generators 3 and 7)

xx .. . . x x .. .. x x xx .. x x x x .. x x x x x x…….……. Reverse HeadReverse Head

11 10 9 8 7 6 5 4 3 2 1 011 10 9 8 7 6 5 4 3 2 1 0

.. x x x x . . x x x x .. . . x x .. .. x x

Duality in LP and IPDuality in LP and IPThe Value Function of an LPThe Value Function of an LP

Minimise xMinimise x22

subject to: 2xsubject to: 2x11 + x + x2 2 >= b >= b11

5x5x11 + 2x + 2x22 <= b <= b22

-x-x11 + x + x22 >= b >= b33

xx11 , x , x22 >= 0 >= 0

Value Function of LPValue Function of LP is is Max( Max( 5b5b11 - - 2b2b2 2 , 1/3( b, 1/3( b11 + 2b + 2b33) , b) , b33))

If bIf b1 1 = 13, b= 13, b2 2 = 30, b= 30, b3 3 = 5= 5 we have Max( 5, 7we have Max( 5, 722/3 /3 , 5 ) = 7, 5 ) = 722

/3 /3

,,

Consistency TesterConsistency Tester is is Max(Max( 2b2b11 – b – b22 , -b , -b22 , -b , -b22 + 2b + 2b33 ) <= 0 ) <= 0 giving Max( -4, -30, -20) <= 0 .giving Max( -4, -30, -20) <= 0 .

(5, -2, 0), (1/3, 0, 2/3), (0, 0, 1) (5, -2, 0), (1/3, 0, 2/3), (0, 0, 1) are vertices of Dual Polytope .are vertices of Dual Polytope .

They give They give marginal marginal rates of change (shadow prices) of optimal objective with rates of change (shadow prices) of optimal objective with

respect to brespect to b11, b, b22, b, b3 3 ..

(5, -2,, 0), (1/3, 0, 2/3), (0, 0, 1) (5, -2,, 0), (1/3, 0, 2/3), (0, 0, 1) are extreme rays of Dual Polytope .are extreme rays of Dual Polytope .

What are the corresponding quantities for an IP ?What are the corresponding quantities for an IP ?

LP SolutionLP Solution

9 . . . . 9 . . . . Min xMin x22

cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30


7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0 xx22

6 . . . . .6 . . . . .

5 . . . . .5 . . . . .

0 1 2 3 4 0 1 2 3 4 xx11


9 9 Optimal IP Solution (2 , 9)Optimal IP Solution (2 , 9) .. . . . . Min xMin x22

cc33 st 2xst 2x11+ x+ x22 >= 13 >= 13

88 . . . . c1c1 . . . . 5x5x11 + 2x + 2x22 <= 30 <= 30


7 . . . . 7 . . . . cc2 2 . . x x11 , x , x2 2 >= 0 >= 0

xx22

6 . . . . .6 . . . . .

5 . . . . .5 . . . . .

0 1 2 3 4 0 1 2 3 4 xx11

Duality in LP and IPDuality in LP and IPThe Value Function of an IPThe Value Function of an IP

Minimise xMinimise x22

subject to: 2xsubject to: 2x11 + x + x2 2 >= b >= b11

5x5x11 + 2x + 2x22 <= b <= b22

-x-x11 + x + x22 >= b >= b33

xx11 , x , x22 >= 0 and integer >= 0 and integer

Value Function of IPValue Function of IP is is

MaxMax( ( 5b5b11 - - 2b2b2 2 ,,┌┌1/3( b1/3( b11 + 2b + 2b33) )

┐┐ , b , b3 3 , , bb11+ 2 + 2 ┌┌ 1/5 (-b 1/5 (-b22+ 2+ 2 ┌┌

1/3(b1/3(b11 + 2b + 2b33) ) ┐ ┐ )) ┐ ┐ ) )

This is known as a This is known as a Gomory Function.Gomory Function.

The component expressions are known as The component expressions are known as ChvChvάάtal Functions tal Functions ..

Consistency Tester Consistency Tester same as for LP (in this example) same as for LP (in this example)

Gomory and ChvGomory and Chváátal Functionstal Functions

Max( 5bMax( 5b11-2b-2b22, , ┌┌1/3(b1/3(b11 + 2b + 2b33) )

┐┐, b, b3 3 , b, b11+ 2 + 2 ┌┌1/5 (-b1/5 (-b22+ 2+ 2 ┌┌1/3(b1/3(b11 + 2b + 2b33) )

┐┐ ) ) ┐┐ ) )

If bIf b11=13, b=13, b22=30, b=30, b33=5 we have Max(5,8,5,9)=9=5 we have Max(5,8,5,9)=9

ChvChváátal Function tal Function bb11+ 2 + 2 ┌┌1/5 (-b1/5 (-b22+ 2+ 2 ┌┌1/3(b1/3(b11 + 2b + 2b33) )

┐ ┐ )) ┐┐ determinesdetermines the optimum.the optimum.

LP Relaxation is LP Relaxation is 19/15 b19/15 b11 - 2/5 b - 2/5 b22 +8/15 b +8/15 b22

(19/15, -2/5, 8/15)(19/15, -2/5, 8/15) is an interior point of dual polytope but is an interior point of dual polytope but

(5, -2, 0)(5, -2, 0) and and (1/3, 0, 2/3)(1/3, 0, 2/3) are vertices of dual corresponding to possible are vertices of dual corresponding to possible LP optima (for different bLP optima (for different b i i ))

Why are valuations on discrete Why are valuations on discrete resources of interest ?resources of interest ?

Allocation of Fixed CostsAllocation of Fixed Costs

Maximise Maximise ∑∑jj p pii x xi i - f y- f y

stst xxi i - D- Dii y y <= 0 <= 0 for all Ifor all I

y y εε {0,1} depending on whether facility built. {0,1} depending on whether facility built.f f is is fixed fixed cost.cost.

xxi i is level of service provided to i (up to level Dis level of service provided to i (up to level Dii ) )ppi i is unit profit to is unit profit to i.i.

A ‘dual value’ A ‘dual value’ vvii on on xxi i - D- Dii y y <= 0 <= 0 would result inwould result in

MaximiseMaximise ∑ ∑jj (p (pii – v – vi i ) x) xi i - (f – (∑- (f – (∑jj D D i i v v ii) y) y

Ie an allocation of the fixed cost back to the ‘consumers’Ie an allocation of the fixed cost back to the ‘consumers’

A Representation for ChvA Representation for Chvátal átal FunctionsFunctions

bb1 1 bb3 3 -- bb22

1 21 2

Multiply and add Multiply and add

on arcs 1on arcs 1

1 1

Divide and round Divide and round

up on nodesup on nodes

22

2 2

GivingGiving b b11+ 2 + 2 ┌┌1/5( -b1/5( -b22+ 2+ 2 ┌┌

1/3( b1/3( b11 + 2b + 2b33) ) ┐┐ ) ) ┐ ┐

LP Relaxation isLP Relaxation is 19/15 b19/15 b11 - 2/5 b - 2/5 b22 +8/15 b +8/15 b33

3

5

1

Simplifications sometimes possibleSimplifications sometimes possible

┌ ┌ 22//7 7 ┌┌ 77//33 nn

┐┐

┐┐ ≡ ≡

┌┌ 22//33 nn ┐┐

ButBut ┌ ┌ 77//3 3 ┌┌ 22//77 nn

┐┐

┐┐ ≠ ≠

┌┌ 22//33 nn ┐ ┐

eg eg nn = 1 = 1

┌ ┌ 11//3 3

┌ ┌ 55//66 nn ┐┐

┐┐

≡ ≡ ┌┌ 55//1818 nn

┐┐

ButBut ┌ ┌ 22//3 3

┌ ┌ 55//66 nn ┐┐

┐┐

≠ ≠ ┌┌ 55//99 nn

┐ ┐ eg eg nn = 5 = 5

Is there a Normal Form ?Is there a Normal Form ?

Properties of ChvProperties of Chvátal Functionsátal Functions

They involve non-negative linear combinations (with possibly negative They involve non-negative linear combinations (with possibly negative coefficients on the arguments) and nested integer round-up.coefficients on the arguments) and nested integer round-up.

They obey the triangle inequality.They obey the triangle inequality.

They are shift-periodic ie value is increased in cyclic pattern with increases They are shift-periodic ie value is increased in cyclic pattern with increases in value of arguments.in value of arguments.

They take the place of inequalities to define non-polyhedral integer They take the place of inequalities to define non-polyhedral integer monoids.monoids.

The Triangle InequalityThe Triangle Inequality

┌┌

aa┐┐

+ + ┌┌

bb┐┐

>= >= ┌┌

a + ba + b┐┐

Hence of value in defining Hence of value in defining Discrete MetricsDiscrete Metrics

A Shift Periodic ChvA Shift Periodic Chvátal Function of átal Function of one argumentone argument

┌ ┌ ½ ( x + 3 ½ ( x + 3 ┌┌ x /9 x /9 ┐ ┐ ) ) ┐ ┐ is is (9, 6) Shift Periodic(9, 6) Shift Periodic. . 22/3/3 is ‘long-run is ‘long-run marginal value’marginal value’

1414

1313

1212

1111

1010

99

88

77

66

55

44

33

22

1 1 .. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ---0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 --- x x


The integer lattice within the polytope -2x + 7y >= 0The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0x – 3y >= 0A Polyhedral MonoidA Polyhedral Monoid

44 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .33 . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .22 . . . . . . . . . . . . .. .. . . . . . . . . . . . . . .1 1 . . . . . . .. . . . . . . . . . . . ……. . . . . . . . . . . . …….00 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Projection: A Non-Polyhedral Monoid (Generators 3 and 7)Projection: A Non-Polyhedral Monoid (Generators 3 and 7) x x .. . . x x .. .. x x x x .. x x x x . . x x x x x x …….……. Defined by Defined by ┌┌-x /3-x /3┐┐ + + ┌┌2x /72x /7┐┐ < = 0 < = 0

FinallyFinally

We should be Optimising We should be Optimising Chvátal Chvátal Functions Functions overover Integer Monoids Integer Monoids

ReferencesReferences CE Blair and RG Jeroslow, The value function of an integerCE Blair and RG Jeroslow, The value function of an integer programmeprogramme, , Mathematical Mathematical

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The Problem With Integer Programming

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